0. Functions

Suppose a quantity described by the function y(t) = y₀eᵏᵗ, where t is measured in years, has a doubling time of 20 years. Find the rate constant k.

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~\(\text{hr}\)$, which means its population is governed by the function $p\(\left\)(t\(\right\))=150\(\cdot{2^{\frac{t}{12}\)}}$, where $t$ is the number of hours after the first observation.

How long does it take the population to triple in size?

Projection sensitivity

According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.

b. Suppose the actual growth rate is instead 0.7%. What are the resulting doubling time and projected 2050 population?

Projection sensitivity

According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.

a. Based on these figures, find the doubling time and the projected population in 2050. Assume the growth rate remains constant.

Solve each equation.

$7^{y-3}=50$

27–30. Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.

Crime rate The homicide rate decreases at a rate of 3%/yr in a city that had 800 homicides/yr in 2018. At this rate, when will the homicide rate reach 600 homicides/yr?

Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.

b. After how many years is the value of the machine 10% of its original value?

Population growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now.

b. Find the doubling time of the population.

After the introduction of foxes on an island, the number of rabbits on the island decreases by 4.5% per month. If y(t) equals the number of rabbits on the island t months after foxes were introduced, find the rate constant k for the exponential decay function y(t) = y₀eᵏᵗ.

15–20. Designing exponential growth functions Complete the following steps for the given situation.

a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.

b. Answer the accompanying question.

Rising costs Between 2010 and 2016, the average rate of inflation was about 1.6%/yr. If a cart of groceries cost \$100 in 2010, what will it cost in 2025, assuming the rate of inflation remains constant at 1.6%?

Savings account A savings account advertises an annual percentage yield (APY) of 5.4%, which means that the balance in the account increases at an annual growth rate of 5.4%/yr.

b. What is the doubling time of the balance?

Changing bases Convert the following expressions to the indicated base.

$\ln \mid x\mid$ using base 5

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~\(\text{hr}\)$, which means its population is governed by the function $p\(\left\)(t\(\right\))=150\(\cdot{2^{\frac{t}{12}\)}}$, where $t$ is the number of hours after the first observation.

How long does it take the population to reach $10,000$?

Tripling time A quantity increases according to the exponential function y(t) = y₀eᵏᵗ. What is the tripling time for the quantity? What is the time required for the quantity to increase p-fold?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If the rate constant of an exponential growth function is increased, its doubling time is decreased.